Parity-Resolved Quantum Capacitance and Quantum Inductance in Topological, Trivial, and Normal Nanowire Interferometers

Abstract

Quantum-capacitance measurements convert the curvature of a quantum-dot energy in a flux-threaded nanowire loop into fast parity-sensitive signals and, therefore, have become a promising readout tool for Majorana devices. However, Majorana-like quantum-capacitance responses can also arise from topologically trivial Andreev bound states, making capacitance alone insufficient to identify a topological phase. To analyze this problem, we consider a quantum dot coupled to both ends of four nanowire realizations: a topological nanowire hosting Majorana bound states, non-topological superconducting nanowires hosting one or two Andreev bound states, and a fully normal nanowire. Motivated by proposals to use quantum inductance as an additional phase-sensitive probe, we compute both the parity-resolved quantum capacitance $C_\mathrm{Q}$ and inverse quantum inductance $L_\mathrm{Q}^{-1}$ as functions of the magnetic flux by exact diagonalization. We show that signatures associated with zero-energy Majorana bound states, such as $h/e$ periodicity and an $h/(2e)$ flux shift between even and odd parity sectors in $C_\mathrm{Q}$ and $L_\mathrm{Q}^{-1}$, are not sufficient indicators of topological superconductivity. In certain realistic parameter regimes, similar Majorana-like behavior can arise from a trivial Andreev bound state and even from a purely normal nanowire. By contrast, two nearly zero-energy Andreev bound states can generate a pronounced $h/(2 e)$-periodic component associated with charge-$2e$ transfer, providing a clear non-Majorana signature. A low-energy projection shows that the Majorana, single-Andreev-state, and normal cases can be mapped onto the same minimal low-energy model explaining their similar flux-dependent responses despite their different physical origins.